Distortion of the hyperbolic robin capacity under a conformal mapping and extremal configurations

B. Dittmar, A. Y. Solynin

Research output: Contribution to journalArticlepeer-review

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Abstract

This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205-218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity δ h (A, Ω) of the boundary set A ⊂ ∂Ω under a conformal mapping of a domain Ω ⊂ U into the unit disk U. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality cap hf(A) ≥ δh (A,Ω) (*) is sharp in the class of conformal mappings f: Ω → U such that f(∂U) = ∂U. Here cap h f(A) is the hyperbolic capacity of a compact set f(A) ⊂ U. We give some examples demonstrating properties of functions which realize the case of equality in relation (*).

Original languageEnglish
Pages (from-to)3058-3069
Number of pages12
JournalJournal of Mathematical Sciences
Volume110
Issue number6
DOIs
StatePublished - 2002

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